IMathAS-libretexts

Properties of Rational Functions, The Horizontal Asymptote
If f(x) has a HORIZONTAL ASYMPTOTE $\displaystyle {y}={a}$ , then as the input values increase or decrease without bound, the output values will approach $\displaystyle {a}$ If the highest degree term in $\displaystyle {p}{\left({x}\right)}$ is greater than the highest degree term in $\displaystyle {q}{\left({x}\right)}$ there will be more than one Horizontal Asymptote The Horizontal Asymptote is a guiding line for the function as the input values increase or decrease without bound If the highest degree term of $\displaystyle {p}{\left({x}\right)}$ is the same as the highest degree term of $\displaystyle {q}{\left({x}\right)}$ then the Horizontal Asymptote is $\displaystyle {x}={0}$ If $\displaystyle {q}{\left({x}\right)}$ has a higher degree term than $\displaystyle {p}{\left({x}\right)}$ then the Horizontal Asymptote is $\displaystyle {y}={0}$

Properties of Rational Functions, The Horizontal Asymptote

Given the Rational Function $\displaystyle {f{{\left({x}\right)}}}=\frac{{{p}{\left({x}\right)}}}{{{q}{\left({x}\right)}}}$ , which of the following statements are true? Check all that apply.

If f(x) has a HORIZONTAL ASYMPTOTE $\displaystyle {y}={a}$ , then as the input values increase or decrease without bound, the output values will approach $\displaystyle {a}$
If the highest degree term in $\displaystyle {p}{\left({x}\right)}$ is greater than the highest degree term in $\displaystyle {q}{\left({x}\right)}$ there will be more than one Horizontal Asymptote
The Horizontal Asymptote is a guiding line for the function as the input values increase or decrease without bound
If the highest degree term of $\displaystyle {p}{\left({x}\right)}$ is the same as the highest degree term of $\displaystyle {q}{\left({x}\right)}$ then the Horizontal Asymptote is $\displaystyle {x}={0}$
If $\displaystyle {q}{\left({x}\right)}$ has a higher degree term than $\displaystyle {p}{\left({x}\right)}$ then the Horizontal Asymptote is $\displaystyle {y}={0}$

License